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Unformatted text preview: i Numbers and Polynomials Notes for a course, Third edition Kermit Sigmon Department of Mathematics University of Florida, Gainesville Copyright c circlecopyrt 1990, 1996 by Kermit Sigmon c circlecopyrt 1999 by Department of Mathematics Department of Mathematics University of Florida Gainesville, FL 32611 revised June 2001, corrections Nov 2005 ii Preface to Second Edition The mathematics you will learn in this course is part of what every mathematically mature person should know. The course is designed neither for learning computational skills nor for learning many facts about number systems and polynomials over these number systems. It is rather to provide you with the opportunity to examine the structure of these systems and to learn the art of careful mathematical reasoning. It is expected that this experience will help you in the linear algebra (MAS 4105) and abstract algebra (MAS 4301) courses which most of you will subsequently take. You are expected to work through the notes, proving the theorems and working the ex ercises. Collaboration between class members in this regard is fine, indeed encouraged, but keep in mind that independent work builds selfconfidence. You should work ahead to the extent that you have worked through the material prior to its discussion in class. In this way, you can compare your work with that discussed in class and develop a confidence that you can independently attack a question. This is a doityourself course in which you are expected to take your turn presenting your work to the class and to actively participate in class discussion. To give you this opportunity for independent discovery, the notes contain neither proofs of the theorems, nor solutions to the exercises. It is important, therefore, for you to keep a carefully organized record of such obtained from your work as modified after class discussion; a ringbinder is suggested for this. The process of carefully rewriting your notes after class discussion is an important part of the learning process in the course. Answers to exercises marked with an asterisk, , are at the end of the notes. Enjoy yourself! Kermit Sigmon Department of Mathematics University of Florida (596) Thanks go to all the instructors of the course whose many suggestions which have improved these notes. These notes were typeset using T E X. Thanks go to Jean Larson for creating the figures in Chapter 6 with T E X. These notes were revised in 1999. iii Introduction We adopt the viewpoint that the real number system is known. More specifically, we will assume the existence of a set R , the real numbers, equipped with two binary operations + and satisfying certain axioms and an order relation &lt; satisfying further axioms. You are to deduce (prove) properties of the real numbers from these axioms and these axioms alone....
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 Spring '08
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